Ind. Eng. Chem. Res. 1996, 35, 2993-2998
2993
Polymer-Induced Turbulent Drag Reduction Hyoung J. Choi† and Myung S. Jhon*,‡ Department of Polymer Science and Engineering, Inha University, Inchon, 402-751, Korea, and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Turbulent drag reduction induced by dilute solutions of both water-soluble poly(ethylene oxide) (PEO) and oil-soluble polyisobutylene (PIB) under turbulent flow in a rotating disk apparatus was investigated. The concentration dependence of drag reduction for these systems was shown to obey an empirical drag reduction equation which has been previously applied to describe flow in circular pipes. Linear correlation between polymer concentration and drag reduction for different molecular weights of PEO and PIB was also obtained. The intrinsic concentration [C] was found to be very useful in normalizing the drag reduction data for different types of polymers, polymer molecular weights, concentrations, and rotational disk velocities. Explanation of the drag reduction phenomenon on a molecular level can be obtained from intrinsic drag reduction and [C]. Introduction Under certain conditions of turbulent flow, the phenomenon in which the drag of a dilute polymer solution is drastically reduced by even minute amounts of suitable additives has been widely investigated. This phenomenon implies that polymer solutions undergoing flow in a pipe require a lower pressure gradient to maintain the same flow rate. A higher flow rate would be obtained for the same pressure gradient if such an additive was used. The various parameters such as polymer concentration, polymer molecular weight, temperature, Reynolds number, and solvent quality are known to be the important factors of drag reduction (DR). The most effective drag-reducing polymers, in general, possess a linear flexible structure and a very high molecular weight. Among the drag-reducing polymers, high molecular weight water-soluble poly(ethylene oxide) (PEO) and oil-soluble polyisobutylene (PIB) were selected for this study. While PEO has been widely used as a drag reducer in aqueous systems, PIB is known to be the only polymer which has received a certain degree of acceptance as a drag-reducing additive for crude oil (Burger et al., 1980). However, since PIB exhibits low stability against mechanical degradation, its drag reduction efficiency degrades rapidly with time. Dschagarowa and Mennig (1977) studied the concentration dependence and the flow rate dependence of PIB, varying the molecular weights, for flow in pipes and obtained a universal drag reduction curve from the intrinsic concentration and the intrinsic drag reduction. They changed solvents to observe the dependence of drag reduction on the intrinsic viscosity or the conformation of PIB and investigated the additional effects of polystyrene in a mixture with PIB in toluene. Although the mechanism of drag reduction has been the subject of extensive research, a complete and satisfactory explanation has not been reported. An understanding of turbulent drag reduction in dilute polymer solutions requires the investigation not only of the * To whom correspondence should be addresssed. Phone: (412) 268-2233. Fax: (412) 268-7139. E-mail: mj3a@ andrew.cmu.edu. † Inha University. ‡ Carnegie Mellon University.
S0888-5885(95)00748-2 CCC: $12.00
phenomena associated with turbulence itself but also of the dynamics of macromolecules in dilute solutions. On the basis of Oldroyd’s theory of slip-at-the-wall (Oldroyd, 1948), Toms (1948) originally proposed the idea of a shear thinning layer at the wall having an extremely low viscosity. However, the rheograms of drag-reducing polymer solutions demonstrate that they were, in fact, not shear thinning. Considering the viscoelastic behavior of drag-reducing polymer solutions near a wall, Ruckenstein (1973) proposed that drag reduction is due to two effects of viscoelasticity: (1) Using a Maxwell model as the constitutive equation for a viscoelastic fluid, he showed that the instantaneous shear stress at the wall is smaller in the viscoelastic fluid than in a corresponding Newtonian fluid. (2) The replacement of the elements of liquid following short paths along the wall takes place as a result of turbulent fluctuations. In order to be replaced by other elements, an element moving along the wall must first relax its elastic stresses to enable viscous deformations required for its replacement to occur. This introduces a delay in the replacement process as compared to a Newtonian fluid. As the instantaneous shear stress at the wall decreases for increasing contact times with the wall, the average shear stress at the wall decreases. In addition, de Gennes (1990) introduced an elastic theory of drag reduction to discuss the properties of homogeneous, isotropic three-dimensional turbulence in the presence of polymer additives without any wall effect. The central idea of this “cascade theory,” limited to linear flexible chains in a good solvent, is that polymer effects on small scales are not described by the viscosity but by the elastic modulus. The importance of an elastic property to describe the mechanism of drag reduction was also examined by Armstrong and Jhon (1984). Adopting a simple model to study both the turbulence and dissolved polymer molecules, they related the molecular dissipation to friction factors by constructing a self-consistent method. For polymer molecules they used a variant of the dumbbell model. The dumbbell model represents a polymer molecule dissolved in a solvent by considering the polymer molecule as two spherical beads connected together by a central spring, immersed in an otherwise Newtonian fluid. Turbulence was also modeled by keeping a kinetic energy budget on the overall flow. They found that a polymer molecule grows by a factor of 10 or more from its equilibrium conformation. © 1996 American Chemical Society
2994 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
In this paper, we will not discuss the theoretical aspect of the origin of turbulent drag reduction but will focus on some empirical correlation among the molecular parameters for both water-soluble PEO and oilsoluble PIB using a rotating disk apparatus (RDA). It is well known that various areas of application for the drag reduction phenomenon can be found both in tube flow and in rotating disk flow. Tube flow consists of various applications, such as pipeline flow (Burger et al., 1980), fire fighting (Fabula, 1971), water supply, irrigation systems (Sellin et al., 1982), hydraulic transportation of solid particle suspensions (Golda, 1986), district cooling and heating circulation systems (Sellin et al., 1982), sewers (Sellin, 1978), and treatment of blood circulation diseases (Greene et al., 1980). Another important application for this phenomenon is in the case of external flow, such as in the case of shipping industries interested in increased velocity and fuel savings. Significant efforts by the Navy and the marine industry have targeted the utilization of drag reducers for ships and submersibles (Vogel and Patterson, 1964; Canham et al., 1971). The rotating disk system investigated in our paper is used to describe the external flow that includes the flow over flat plates as well as the flow around submerged objects. One studies typical friction drag for an internal flow, whereas the other studies the total drag (friction plus form drag) for an external flow. The drag reduction is known to be related only to the friction drag. Therefore, in order to study the total drag reduction, the rotating disk system is adopted in this study. Because of this difference of the friction drag and the total drag between the tube flow (inner flow) and the rotating disk flow (outer flow), a maximum of 80% of the drag reduction can be obtained from the tube flow. However, the rotating disk flow generally produces about 50% of the maximum drag reduction. In addition, as one of the external flows, Couette systems have also been investigated (Kulicke, 1986; Tong et al., 1990). Tong et al. (1990) recently studied the anisotropy of turbulent drag reduction in a concentric cylindrical cell using the novel technique of photon-correlation homodyne spectroscopy. It is well known that drag reduction is strongly influenced by the molecular parameters of the dissolved polymer. Therefore, we concentrated our efforts on the characterization of dilute polymer solutions. First, the concentration dependence of drag reduction was investigated. The efficiency of drag-reducing polymer additives, based upon a unit concentration at infinite dilution, was then determined by using a characteristic parameter, DRmax/[C]. Through these analyses, the universal curve established by previous researchers (Little, 1969; Virk et al., 1967) for tube flow systems was also obtained for the family of both PEO and PIB using an RDA. Virk et al. (1967) investigated the relationship of the relative drag reduction/concentration versus the concentration for various molecular weights of PEO in water flowing through a pipe. They observed that the extent of drag reduction induced by a homologous series of polymers in a given pipe is a universal function of concentration, flow rate, and molecular weight. Improving the form of Virk’s drag reduction equation, Little (1969) and other researchers studied the concentration dependence of drag reduction and found that a linear relationship exists between DRmax/[C] and the molecular weight of the polymer. In addition, [C] was found to be extremely useful in normalizing the
Table 1. Properties of Polymers Used in the Experimentsa polymer grade
Mv
polymer grade
Mv
PEO136E PEO343 PEO344 PEO345
290 000 370 000 1 200 000 2 170 000
PIBL-80 PIBL-100 PIBL-120 PIBL-140
990 000 1 200 000 1 600 000 2 100 000
a
Taken from the manufacturer’s information.
drag reduction data of different molecular weight compounds into one homologous series of PEO. When (DR/ C)/[DR] and C/[C] are defined as β and R, respectively, it is possible to obtain a universal curve represented by β ) 1/(R + 1) for the PEO family. This result suggests that a general universal curve can be written as β ) 1/(R + K), where the constant K is characteristic of a particular polymer family in a given solvent and does not depend on the molecular weight or the flow geometry. In the present study, we investigate whether the same relations can be obtained for PEO-water and PIB-kerosene solutions in an RDA. Experimental Section Four PIBs with different molecular weights (Vistanex MM Grades L-80, L-100, L-120, and L-140), obtained from Exxon Chemical Americas (Choi, 1987), were used as drag-reducing additives with kerosene as a solvent. The purity of PIB is about 99.3 wt % (ash, 0.3 wt %; volatiles, 0.3 wt %; stabilizer, 0.1 wt %), and its molecular weight distribution is quite polydisperse. Vistanex PIBs, highly paraffinic hydrocarbon polymers, are composed of terminally unsaturated long straightchain molecules and have properties of being light in color, odorless, tasteless, and nontoxic. In addition, four different molecular weight PEO samples, purchased from Scientific Polymer Products in powder form (98 wt % through 10 mesh), with viscosity-average molecular weights ranging from 2.9 × 105 to 2.2 × 106 (weightaverage molecular weights ranging from 4.0 × 105 to 5.0 × 106) were used. The properties of the polymers are given in Table 1. Polymer solutions were made by dissolving an appropriate amount of either PEO in distilled water or PIB in kerosene. Stock solutions (0.5 wt % concentration) were initially prepared and then diluted to the required polymer concentration by carefully injecting measured quantities of stock solution directly into the turbulent flow field. In the case of the PEO stock solution, 1 wt % isopropyl alcohol was added to prevent chemical degradation of the polymer. Mild agitation was applied to both polymer systems to reduce mechanical degradation induced by stirring. A schematic diagram of the RDA is shown in Figure 1. It consists of a stainless steel disk whose dimensions are 10.1 cm diameter × 0.32 cm thickness, enclosed in a cylindrical thermostatically controlled container, which is made of stainless steel and whose dimensions are 16.3 cm i.d. (inner diameter) × 5.5 cm height. The volume of solution required to fill the entire container is about 1020 cm3. The rotational velocity of the disk was maintained constant, using a speed controller (Cole Parmer Master Servodyne Unit) and a dc-varying speed motor, and the variable torque was measured by a multimeter. The temperature of the system was maintained at 25.0 ( 0.5 °C by a constant temperature circulating apparatus, and the rotational velocity of the disk was measured by a digital tachometer.
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2995
Figure 1. Schematic diagram of a rotating disk apparatus. (1) speed controller, (2) thermocouple, (3) motor, (4) solution container, (5) water bath, (6) water-circulating system, and (7) thermometer.
A flow visualization experiment to determine whether the fluid was uniformly mixed after the addition of the stock solution was employed with the same apparatus used by Yang et al. (1991). They observed complete mixing within several seconds and obtained reproducible results for each run. However, experimental error could have occurred during the early time period when uniform mixing was not yet achieved. The drag reduction efficiency was obtained by first measuring the torque required to rotate the disk at a given speed in the pure solvent. By measuring the corresponding torque required to attain the same speed in the solvent with polymer added, the percentage reduction is then calculated as follows:
DR (%) )
T0 - Tp × 100 T0
(1)
where T0 is the torque in the pure solvent and Tp is the torque required when the polymer is added, that is, the torque in the dilute polymer solution. The rotational Reynolds number NRe is defined as
NRe ) Fr2ω/µ
(2)
where F and µ are the density and the viscosity of the fluid, r is the radius of the disk, and ω is the rotational speed of the disk. Using the RDA, turbulence is produced for NRe > 3 × 105 or equivalently 1050 rpm for the rotational disk velocity (Schlichting, 1979). Since the drag reduction phenomenon for the polymer occurs only in the turbulent region, all RDA measurements in this study were taken much above 1050 rpm. No drag reduction was observed below 1050 rpm (Kim et al., 1993). In addition, an Ubbelohde viscometer was used to measure the intrinsic viscosity [η] at 30.0 ( 0.05 °C in deionized water for PEO and in toluene for PIB using the Huggins equation
ηsp/C0 ≡ (ηp - η0)/η0C0 ) [η] + k′[η]2C0 where C0 is the polymer concentration (g/dL), η0 is the solvent viscosity, ηp is the viscosity of the polymer solution, and ηsp is the specific viscosity. Viscosity-average molecular weights (Mv) of PEO and PIB were then estimated using the following Mark-Houwink equation: [η] ) RMbv where R ) 1.25 × 10-4 and b ) 0.78 for PEO, and R ) 0.02 and b ) 0.67 for PIB. Results and Discussion To find an empirical relationship between the drag reduction and the polymer solution properties, Virk et
al. (1967) experimentally studied the drag reduction caused by dilute PEO solutions in turbulent pipe flow and showed that the extent of drag reduction induced by a homologous series of polymers is a universal function of concentration, flow rate, and the molecular weight for a given geometry. Little (1969) suggested that the critical polymer concentration, defined as the concentration where random coils begin to touch, might be used to normalize the drag reduction data since the same critical concentration appeared to produce the same degree of drag reduction, irrespective of the molecular weight of the polymer. We will first introduce the three-parameter empirical relationship between the drag reduction (DR) and the concentration (C) to provide a universal correlation for drag reduction data. We first cast DR in PadÄ e form as: DR ) P(C)/Q(C) where P(C) and Q(C) are polynomials of C. Since we have only three easily observable relationships, i.e.
lim DR ) 0
(3)
Cf0
lim (DR/C) ) [DR]
(4)
Cf0
and
lim DR ) DRmax
(5)
Cf∞
where DRmax is the maximum percent drag reduction for a given polymer solution. [DR] is the intrinsic drag reduction which implies a measure for drag-reducing efficiency of the initial increments of polymer, and C is the polymer concentration (wppm, parts per million based on weight). We can model DR in the following simplified form:
DR )
a0 + a1C b0 + b1C
(6)
From eqs 3-5, we can easily determine a0, a1, b0, and b1, and (by setting b0 t 1)
a0 ) 0, a1 ) [DR], and b1 )
[DR] 1 (7) ≡ DRmax [C]
Therefore, an empirical relationship can be written as
DR )
C[DR] 1 + C/[C]
or
DR/C 1 ) [DR] 1 + C/[C]
(8)
For experimental purposes, the two important quantities, previously reported by Little (1969), are intrinsic drag reduction [DR] and intrinsic concentration [C] defined in eqs 4 and 7. From eq 7, [DR] can be expressed again as DRmax/[C] and becomes a measure of the efficiency of the polymer additives on a unit concentration basis at infinite dilution. Hunston and Zakin (1980) found that for various polymer-solvent systems, more efficient materials have a larger DRmax and a smaller [C], and both DRmax and [C] are more favorable in a good solvent. To interpret drag reduction data, one can rewrite eq 8 in the following form:
[C] C C ) + DR DRmax DRmax
(9)
2996 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 2. (a, top) Drag reduction (DR) of PEO versus the concentration of polymers of various molecular weights at 2800 rpm. (b, bottom) DR versus the concentration of PIB of different molecular weights at 1800 rpm.
Although the above derivation is straightforward, it may be the most systematic modeling of the universal correlation for drag reduction. In this paper, we have introduced an additional parameter, b0 ) K, to improve Little’s universal correlation as shown:
1 DR/C ) [DR] K + C/[C]
(10)
To improve the fit of the drag reduction data, we use the Pade´ form with several additional parameters rather than a Taylor expansion. Since drag reduction is caused by the sum of the contributions from individual polymer molecules, increasing concentrations of polymer solutions increase the amount of drag reduction. Figure 2a shows the dependence of the percent drag reduction of four different molecular weights of PEO as a function of polymer concentration up to 250 wppm at the RDA rotational velocity of 2800 rpm. The dependence of the percent drag reduction on four different molecular weights of PIB as a function of polymer concentration up to 1000 wppm at the rotational disk velocity of 1800 rpm was also determined as shown in Figure 2b. From Figure 2 the concentrations having the maximum drag reduction at each different molecular weight can be obtained and are called optimum concentrations. The optimum concentration decreases as the molecular weight of the polymer increases for both PEO and PIB. The values of the maximum drag reduction for different molecular weights of PEO and PIB, obtained from RDA measurements, are used in the analysis. The data
Figure 3. (a, top) Concentration dependence of drag reduction for PEO at 2800 rpm. (b, bottom) Concentration dependence of drag reduction for PIB at 1800 rpm.
clearly indicate that the concentration required for maximum drag reduction decreases with increasing molecular weight. On the other hand, drag reduction becomes a weak function of molecular weight for PEO concentrations above 150 wppm. The maximum in the drag reduction-concentration data in Figure 2 is due to the combination of the two following factors: the drag-reducing property of the solute and the increasing viscosity of the solutions, which becomes increasingly significant at higher concentrations of the polymer. The linear correlation between the polymer concentration and C/DR for four different molecular weights of PEO and PIB in a range of conditions close to the maximum drag reduction is illustrated in parts a and b, respectively, Figure 3, showing that eq 9 can also be applied to drag-reducing polymer-solvent systems with different geometries (e.g., RDA). For a dilute solution, Little et al. (1975) explained that Henry’s law is applicable in the range in which C/[C] values approach 0.01. The parameter DRmax/[C], the Henry’s law constant shown in eq 11, defines the efficiency of the polymer additives on a unit concentration basis at infinite dilution.
lim Cf0
DR C
) lim Cf0
DRmax
DRmax )
C + [C]
(11)
[C]
Only at these intermolecular distances (for dilute solutions) is the percent drag reduction a linear function of polymer concentration. Near the maximum DR condition, the polymer molecules are only a few diam-
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 2997
Figure 4. (a, top) DRmax/[C] versus molecular weight for PEO at 2800 rpm. (b, bottom) DRmax/[C] versus molecular weight for PIB at 1800 rpm.
Figure 5. (a, top) Intrinsic concentration as a nomalizing factor for PEO at 2800 rpm. (b, bottom) Intrinsic concentration as a nomalizing factor for PIB at 1800 rpm.
eters apart, and they are virtually in contact. Equation 11 is valid not only in a turbulent pipe system but also in our RDA. The intercept value at C/DR ) 0 yields the intrinsic concentration [C], and this quantity divided by the intercept at C ) 0 gives DRmax. Figure 4 shows a correlation between the intrinsic drag reduction in eq 11 and the viscosity-average molecular weight for the homologous series of both PEO and PIB. The plot is linear and the intercept values of Mv ) 2.65 × 105 for PEO and Mv ) 9.0 × 105 for PIB obtained from the least squares method provide us with a lower limit point in molecular weight below which no drag reduction takes place. Even though this value for PEO is comparable with that previously reported for the Polyox family, Mv ) 2.46 × 105 in pipe flow system at NRe ) 9000 (Ting and Little, 1973), the cutoff molecular weight was observed to be different for the different tube diameters in a pipe flow system (Little, 1971). Furthermore, [C] was found to be extremely useful in normalizing the drag reduction data of different molecular weight compounds into one homologous series as shown in Figure 5. This figure demonstrates that the intrinsic concentration can be used to normalize data obtained from the RDA. On the other hand, if we define (DR/C)/[DR] as β and C/[C] as R, eq 8 becomes β ) 1/(R + 1), and this figure also indirectly indicates that eq 8 is a universal curve equation for the PEO familywater in an RDA system. However, as shown in Figure 5b for RDA measurements of PIB in kerosene, the universal curve is well represented by β ) 1/(R + 0.4). For toluene-soluble PIB, Dschagarowa and Mennig (1977) obtained the universal curve β ) 1/(R + 0.4) for a capillary tube system. From the similar behavior
observed for both water-soluble PEO and toluenesoluble PIB, we can propose a universal curve for any flow geometry with a single parameter which depends on the polymer-solvent system,
β ) 1/(R + K)
(12)
as derived in eq 10. The constant K is characteristic of a particular polymer family in a given solvent and does not depend on the molecular weight or flow geometry of the system. The value of K is 1 for PEO-water in a pipe flow system (Virk et al., 1967). For PIB-toluene and PIB-kerosene in a pipe flow system (Dschagarowa and Mennig, 1977), K is 0.4. From these results, the constant K (or b0 in eq 6) is found to be a specific parameter inherent to the interaction between a polymer in a given solvent. The low K value for PIBtoluene (or PIB-kerosene), compared with that for PEO-water, comes from the differences in the solubility parameter. The solubility parameters of PIB, PEO, toluene, and water are 15.5, 20.4, 18.2, and 47.9 MPa1/2, respectively. In addition, the approximate solubility parameter of kerosene is about 17, since kerosene is mainly composed of n-dodecane and other chemicals such as trialkyl derivatives of benzene, naphthalene, and 1-n-2-methyl-5,6,7,8-tetrahydronaphthalene. Therefore, both toluene and kerosene are better solvents for PIB, giving a low value for K. Further investigation of this argument is currently underway for various solvents and polymer systems. Figure 6 clearly indicates this universal characteristic for PIB in kerosene, irrespective of the molecular weight and the rotational disk velocity.
2998 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 6. Universal drag reduction curve, (DR/C)/[DR] vs C/[C], for PIB in kerosene.
The empirical equations relating drag reduction to relevant solution properties reduces the amount of time needed in evaluation procedures. The polymer concentration was used to normalize the drag reduction data since the same fraction of the critical concentration appeared to produce the same degree of drag reduction, irrespective of the polymer molecular weight. Conclusions The effect of concentration for a homologous series of PEO and PIB on drag reduction was investigated in a rotating disk apparatus. Increases in the percent drag reduction of polymer solutions were obtained by incremental increases of concentration up to a certain concentration, which is referred to as the optimum concentration, and then the percent drag reduction decreases with concentrations greater than the optimum concentration. Higher molecular weight polymers show a maximum drag reduction at lower concentrations. The intrinsic drag reduction, the intrinsic concentration, and the universal drag reduction curve for PEO in water and for PIB-kerosene in an RDA agree well with those for PIB-toluene and for PEO-water in a pipe system. That is, the universality of a drag-reducing polymer is the same regardless of the types of flow geometry and solvent. We also derived a threeparameter relationship between DR and C, using the Pade´ form (see eq 10), which gives a universal curve. Acknowledgment H.J.C. is grateful to the Korea Science and Engineering Foundation (Project No. 94-02-08-11) for the financial support of this work. Literature Cited Armstrong, R.; Jhon, M. S. A Self-consistent Theoretical Approach to Polymer Induced Turbulent Drag Reduction. Chem. Eng. Commun. 1984, 30, 99. Burger, E. D.; Chorn, L. G.; Perkins, T. K. Studies of Drag Reduction Conducted over a Broad Range of Pipeline Conditions when Flowing Prudhoe Bay Crude Oil. J. Rheol. 1980, 24, 603. Canham, H. J. S.; Catchpole, J. P.; Long, R. F. Boundary Layer Additives to Reduce Ship Resistance. The Naval Architect. J. Rina 1971, 2, 187.
Choi, H. J. Direction of Lateral Migration of a Rigid Sphere in a Homologous Series of Polymer Solutions. Ph.D. Dissertation, Carnegie Mellon University, Pittsburgh, PA, 1987. De Gennes, P. G. Introduction to Polymer Dynamics ; Cambridge University Press: Cambridge, Great Britian, 1990. Dschagarowa, E.; Mennig. G. Influence on Molecular Weight and Molecular Conformation of Polymers on Turbulent Drag Reduction. Rheol. Acta 1977, 16, 309. Fabula, A. G. Fire Fighting Benefits of Polymeric Friction Reduction. Trans. ASME J. Basic Eng. 1971, 93D, 453. Golda, J. Hydraulic Transport of Coal in Pipes with Drag Reducing Additives. Chem. Eng. Commun. 1986, 43, 53. Greene, H. L.; Mostardi, R. F.; Wokes, R. F. Effects of Drag Reducing Polymers on Initiation of Arteriosclerosis. Polym. Eng. Sci. 1980, 20, 499. Hunston, D. L.; Zakin, J. L. Flow-Assisted Degradation in Dilute Polymer Solutions. Polym. Eng. Sci. 1980, 20, 517. Kim, C. B.; Yang, K. S.; Choi. H. J.; Jhon, M. S. Drag Reducing Effects of Polymer Additives on Coal-Water Mixture in Rotating Disk System. KSME J. 1993, 7, 48. Kulicke, W.-M. Unusual Instability Effects Observed in Ionic and Non-ionic Water Soluble Polymers. Makromol. Chem., Macromol. Symp. 1986, 2, 137. Little, R. C. Flow Properties of Polyox Solutions. Ind. Eng. Chem. Fundam. 1969, 8, 557. Little, R. C. Drag Reduction in Capillary Tubes as a Function of Polymer Concentration and Molecular Weight. J. Colloid Interface Sci. 1971, 37, 811. Little, R. C.; Hansen, R. J.; Hunston, D. L.; Kim, O. K.; Patterson, R. L.; Ting, R. Y. The Drag Reduction Phenomenon. Observed Characteristics, Improved Agents and Proposed Mechanisms. Ind. Eng. Chem. Fundam. 1975, 14, 283. Oldroyd, J. G. Suggested Method of Detecting Wall Effects on Turbulent Flow through Tubes. Proceedings of the 1st Congress on Rheology; North-Holland: Amsterdam, 1948; Vol. 2, p 130. Ruckenstein, E. A. Note on the Mechanism of Drag Reduction. J. Appl. Polym. Sci. 1973, 17, 3239. Schlichting, H. Boundary Layer Theory, 7th ed.; McGraw-Hill: New York, 1979. Sellin, R. H. J.; Hoyt, J.; W.; Pollert, J.; Scrivener, O. The Effect of Drag Reducing Additives on Fluid Flows and Their Industrial Applications Part 2: Present Applications and Future Proposals. J. Hydraul. Res. 1982, 20, 235. Sellin, R. H. J. Drag Reduction in Sewers: First Results from a Permanent Installation. J. Hydraul. Res. 1978, 16, 337. Ting, R. Y.; Little, R. C. Characterization of Drag Reduction and Degradation Effects in the Turbulent Pipe Flow of Dilute Polymer Solutions. J. Appl. Polym. Sci. 1973, 17, 3345. Toms, B. A. Some Observations on the Flow of Linear Polymer Solutions through Straight Tubes at Large Reynolds-Numbers. Proceedings of the 1st Congress on Rheology; North-Holland: Amsterdam, 1948; Vol. 2, p 135. Tong , P.; Goldburg, W. I.; Huang, J. S.; Witten, T. A. Anisotropic in Turbulent Drag Reduction. Phys. Rev. Lett. 1990, 65, 2780. Virk, P. S.; Merrill, E. W.; Mickley, H. S.; Smith, K. A.; MolloChristensen, E. L. The Toms Phenomenon-Turbulent Pipe Flow Dilute Polymer Solutions. J. Fluid Mech. 1967, 30, 305. Vogel, W. H.; Patterson, A. M. An Experimental Investigation of the Additives Injected into the Boundary Layer of an Underwater Body. Proceedings of 5th Symposium on Naval Hydrodynamics; ONR-ACR-112, Bergen, 1964, p 975. Yang, K. S.; Choi, H. J.; Kim, C. B.; Jhon, M. S. A Study of Drag Reduction by Polymer Additives in Rotating Disk Geometry. Korean J. Rheol. 1991, 3 (1), 76.
Received for review December 15, 1995 Accepted April 5, 1996X IE9507484
X Abstract published in Advance ACS Abstracts, August 15, 1996.
